We study the robustness of the results of Milevsky and Huang (2018) on the optimal demand for annuities to the choice of the utility function. To do so, we first propose a new way to span the set of all increasing concave utility functions by exploiting a one-to-one correspondence with the set of probability distribution functions. For example, this approach makes it possible to present a five-parameter family of concave utility functions that encompasses a number of standard concave utility functions, e.g., CRRA, CARA and HARA. Second, we develop a novel numerical method to handle the life-cycle model of Yaari (1965) and the annuity equivalent wealth problem for a general utility function. We show that the results of Milevsky and Huang (2018) on the optimal demand for annuities proved in the case of a CRRA and logarithmic utility maximizer hold more generally.